ALGEBRA. 



Since the square of every quantity is 

 positive, a negative quantity has no square 

 root ; the conclusion therefore shews that 

 there are no such numbers as the ques- 

 tion supposes. See BINOMIAL THEO- 

 REM; EQUATIONS, nature of,- SERIES, 

 SURDS, &c. 8cc. 



ALGEBRA, application of to geometry. 

 The first and principal applications of al- 

 gebra were to arithmetical questions and 

 computations, as being the first and most 

 useful science in all the concerns of hu- 

 man life. Afterwards algebra was applied 

 to geometry, and all the other sciences 

 in their turn. The application of algebra 

 to geometry is of two kinds; that which 

 regards the plane or common geometry, 

 and that which respects the higher geo- 

 metry, or the nature of curve lines. 



The first of these, or the application of 

 algebra to common geometry, is concern- 

 ed in the algebraical solution of geome- 

 trical problems, and finding out theorems 

 in geometrical figures, by means of alge- 

 braical investigations or demonstrations. 

 This kind of application has been made 

 from the time of the most early writers on 

 algebra, as Diophantus, Cardan, &c. &c. 

 down to the present times. Some of the 

 best precepts and exercises of this kind 

 of application are to be met with in Sir I. 

 Newton's " Universal Arithmetic," and in 

 Thomas Simpson's " Algebra and Select 

 Exercises." Geometrical problems are 

 commonly resolved more directly and ea- 

 sily by algebra, than by the geometrical 

 analysis, especially by young beginners; 

 but then the synthesis, or construction 

 and demonstration, is most elegant as de- 

 duced from the latter method. Now it 

 commonly happens, that the algebraical 

 solution succeeds best in such problems 

 as respect the sides and other lines in ge- 

 ometrical figures ; and, on the contrary, 

 those problems in which angles are con- 

 cerned are best effected by the geome- 

 trical analysis. Sir Isaac Newton gives 

 among many other remarks on this 



branch. Having any problem proposed, 

 compare together the quantities concern- 

 ed in it; and making no difference be- 

 tween the known andunknown quantities, 

 consider how they depend, or are related 

 to, one another ; that we may perceive 

 what quantities, if they are assumed, will, 

 by proceeding synthetically, give the rest, 

 and that in the simplest manner. And in 

 this comparison, the geometrical figure is 

 to be feigned and constructed at random, 

 as if all the parts were actually known or 

 given, and any other lines drawn, that may 

 appear to conduce to the easier and sim- 

 pler solution of the problem. Having 

 considered the method of computation, 

 and drawn out the scheme, names are 

 then to be given to the quantities enter- 

 ing into the computation, that is, to some 

 few of them, both known andunknown, 

 from which the rest may most naturally 

 and simply be derived or expressed, by 

 means of the geometrical properties of 

 figures, till an equation be obtained, by 

 which the value of the unknown quantity 

 may be derived by the ordinary methods 

 of reduction of equations, when only one 

 unknown quantity is in the notation ; or 

 till as many equations are obtained as 

 there are unknown letters in the notation. 

 For example : suppose it were required 

 to inscribe a square in a given triangle. 

 Let ABC, (Plate Miscellanies, fig. 1.) be 

 the given triangle: and feign DEFGto be 

 the required square : also draw the per- 

 pendicular BP of the triangle, which will 

 be given, as well as all the sides of it. 

 Then, considering that the triangles BAC, 

 BEF are similar, it will be proper to make 

 the notation as follows, viz. making the 

 base A.C=6, the perpendicular BP=/>, 

 and the side of the square DE or EF=a:. 

 Hence then BQ=BP ED=/> x- 

 consequently,by the proportionality of the 

 parts of those two similar triangles, viz. 

 BP : AC :: BQ : EF, it isp : b :: p X : X ; 

 then, multiply extremes and means, &c. 

 there arises p x=b p b x t or b x-{-p x 



=l> p, and x= , , the side of the square 

 b-\-p 



sought ; that is, a fourth proportional to 

 the base and perpendicular, and the sum 

 of the two, taking this sum for the first 

 term, or AC-j-BP : BP :: AC : EF. 



The other branch of the application of 

 algebra to geometry was introduced by 

 Descartes, in his Geometry, which is the 

 new or higher geometry, and respects the 

 nature and property of curve lines. In 

 this branch, the nature of the curve is ex- 

 pressed or denoted by an algebraic equa- 

 tion, which is thus derived : A line is 



